You actually can't see a bullet mainly due to size.
Upgrade it to something bigger and it's another story. For example:
The talk is about debris - you can say that this satellite chunk that misses Lt. Kowalski is a big one. The bigger they are the easier they are to spot...
Another important bit is that in space there is no atmosphere so visibility is much better. Thus you can see much smaller things much farther out than on Earth. This is, BTW, the reason why space-borne telescopes don't have to be as big as the ground-side ones to yield images in much higher resolution and quality.
EDIT: after reading some comments I think It's time to add some info.
First of all it's incorrect to assume all satellites travel at same speed, same altitude and in same direction. For example Hubble's period is about 95 minutes, which would translate to about 9 km/s (32000 km/h). Hubble also, probably, is in sun-synchronous orbit (slightly retrograde) brackets, meaning it travels in a huge pack of other satellites, passes another bunch or two on its way each revolution.
Debris in question came from satellite hit by a missile, which turned into a debris cloud travelling upwards at the speed of 32000 km/h (according to data in the video). However, this is a spy satellite (which means LEO, which means about 400-450 km of distance between the two orbits). Missile strike obviously imparted some of it's speed on the debris itself (added or detracted, depending on the exact orbit) and depending on the direction from the missile came it could significantly change the vectors. And we're talking about MULTIPLE satellites...
In other words: all those factors may represent huge velocity difference as a result. Different inclination, different orbit period, tangential motion... all adds up.
EDIT 2: And by "it's all adding up" i mean this: russians bungled it up enormously by using missile (from POV of orbital safety). Usually satellites are disposed by burning them in the atmosphere (where possible) to avoid the situation exactly as shown in the movie. Go to stuffin.space to see just how chock full of stuff the Earth orbit is. from NASA appraisals majority of that stuff is in 0-100 kg range, meaning pretty heavy projectile. But back to destroyed satellite. Using missile means intercept trajectory, not pursuit (fuel limitations and depending on the location of the interceptor's launch big-to-severe speed penalty). Successful interception means change in speed (it will be significantly reduced) and trajectory of the debris (as mentioned in the movie: "upward"). Since we have around 2300 satellites overhead, with most of them in similar orbit, inclination and period (and other stuff - hopefully I'm using right words), there will be a chain reaction. Each "succesfully" destroyed satellite will create it's own debris cloud, and each time some of the debris will be moving slower than before and more and more "upward" (i.e. away from Earth). Repeat it multiple times and in the end we have this situation:
Shuttle, traveling at his nominal speed, enters the debris field which moves across his trajectory much slower and at an angle. This would be the equivalent - to use Nathan's cool analogy of a car from comments below - of running red light during a traffic jam at a high speed Since the russian satellite almost certainly was moving slower than the shuttle to start with (as it being "probably spy sat", to the tune of the "low-to-mid speed bullet"), now difference in speed is staggering.
As far as the OP question goes: satellite is usually the size of a human being. How far out can you see human being? On Earth it's about 6 km (curvature and atmosphere get in the way), in space it can be much farther (but, admittedly, due to eye limitations as a speck of reflected sunlight and without the ability to gauge distance from it until much bigger).
Disclaimer: always when speed is used with no other qualificators it's meant to be speed relative to Earth